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%  \cl{\bf About the Costa-Hoffman-Meeks Minimal Surfaces}    This is too wide, so let's divide it
 
\cl{\bf              About the Costa-Hoffman-Meeks    }                                   
\cl {\bf                      Minimal Surfaces                   }

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\cl{                               H. Karcher                         }
\lf
The original Costa surface was responsible for the rekindling 
of  interest in minimal surfaces in 1982. It is a minimal 
{\bf embedding} of the 3-punctured square torus. Its planar symmetry 
lines cut this surface into four conformal squares and the two 
straight lines through the saddle are the diagonals of these 
squares. Because of the emphasis on the symmetries, our 
formulas are taken from [K2.]  The Costa-Hoffman-Meeks 
surfaces are generalizations of the Costa surface;  their genus
grows as the dihedral symmetry (controlled by dd) is increased. 
The underlying Riemann surfaces are tesselated by hyperbolic
squares with angles $\pi \over k$, $(k = 2,3, ...)$.


\lf
The Gauss map of such a surface is determined by its
qualitative properties only up to a multiplicative factor cc
which we suggest for the morphing (as in the Chen-Gackstatter
case).  It closes the period (at cc0) by an intermediate value
argument.

\lf
As in Costa's case, the qualitative picture determines the Gauss
map only up to a multiplicative factor. The standard morph
shows the dependence of the surfaces on this factor, closing
the period at cc0.

\lf
[Hoffman, Karcher] Complete embedded minimal surfaces of
finite total curvature. Encyclopaedia of Mathematical Sciences,
vol. 90. Geometry V (Ed. R. Osserman), pp. 5-93

\lf
[K2]  H. Karcher, Construction of minimal surfaces, in ``Surveys in
      Geometry'', Univ. of Tokyo, 1989, and Lecture Notes No. 12,
      SFB 256, Bonn, 1989, pp. 1--96.


\lf
  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

\lf
  [KWH]  H. Karcher, F. Wei, and D. Hoffman,  \hfil\break
       The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993


\lf
[DHKW] U. Dierkes, S. Hildebrand, A. Kuster,  O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991



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